Six-DOF motion testing and motion parameter decoupling method for rotors based on shaft-disk

ABSTRACT

A six-DOF motion testing and motion parameter decoupling method for rotors based on shaft-disk is proposed, which includes a displacement sensor tooling and a precision shaft-disk fixed on the rotor where three measuring points are arranged on the surface of disk to measure the axial motion of the rotor, two measuring points on the shaft to measure the radial motion, and the angle encoder at the shaft shoulder to measure the rotation motion. The tooling guarantees the accuracy of displacement sensors. The fixed coordinate system and the shaft-disk moving coordinate system are set, and the measured values of the displacement sensors and the encoder are represented by vectors to establish the relationship between the six-DOF motion of the shaft-disk axis and the measured values of sensors. Thus, the six-DOF motion of the rotor/shaft-disk can be determined by the measured data.

TECHNICAL FIELD

The invention belongs to the technical field of the rotor motionaccuracy test, and relates to a six-DOF motion testing and motionparameter decoupling method for rotors based on shaft-disk.

BACKGROUND

The actual motion parameters of a rotor are the key indexes to evaluatethe motion and power transmission quality, which directly affect theaccuracy of precision machine tools, instrument turntables, geartransmission devices and other mechanical equipment. The nominal motionof a rotor is a single-DOF rotation, and the other five DOFs are rigidlyconstrained. However, due to the manufacturing errors and elasticdeformation of each part, the actual motion of a rotor presents six-DOFspatial behavior which is necessary to be evaluated by the six-DOFmotion testing and motion parameter decoupling method. In the 1970s, theInternational Institution for Production Engineering Research publisheda unified document on the measurement of the error motion performance ofthe rotary axis which defines the unified terms about the axis of arotor to promote the research of the rotor motion test method. Nowadays,the rotor motion test is widely used in the precision equipment accuracyevaluation, error compensation and fault diagnosis. With the increasingdemand of the accuracy of mechanical equipment, the problem of six-DOFmotion test and its precise decoupling of motion parameters isincreasingly prominent, which has become the main problem of the rotormotion performance evaluation, compensation and fault diagnosis.

For the rotor motion test, one-direction measurement and two-directionmeasurement were applied to test the radial motion of a rotor at first.Thereafter, the method of combining the radial motion test and the axialmotion test was adopted, in which the displacement sensors are installedin three directions of the coordinate system to detect the radial andaxial motion of the standard bar fixed on the rotor. The rotor motion inthe inclination direction can be tested by arranging displacementsensors in multiple sections. The above test method can only test somemotion parameters (radial, axial or inclination) of rotors. In 1992,Lion Company of the United States used a five-point method based on thedouble standard ball and angle encoder to test the actual motion of arotor, which can realize the rotor motion test with six DOFs. However,the machining of the double standard ball is very difficult and theoperating condition of the rotor motion test is very harsh, especiallythe axial test has high requirements on rotor structure and space, whichis difficult to be applied to all rotor motion tests. Another method isto measure the radial motion of two sections of the standard bar by fourdisplacement sensors and the axial motion by one displacement sensor,which can simply measure the six-DOF motion of rotors. However, thistest scheme takes up a large space and the axial displacement sensor isdifficult to be arranged in the closed transmission chain. The testsurface of the bar is cylindrical but not spherical, i.e. the verticalposition from the measuring point of the bar to the rotor axis alwayschanges. Although the six-DOF motion parameters can be decoupled byassuming that the vertical position is constant, the measurementprinciple error is introduced, which makes the decoupled motionparameters imprecise and difficult to meet the motion test requirementsof precision equipment.

SUMMARY

In order to solve the problem of the universality of six-DOF motion testof rotors and that of accurate decoupling of six-DOF motion parametersexisting in the prior art, the present invention proposes a six-DOFmotion testing and motion parameter decoupling method of rotors based onshaft-disk. Instead of the double standard ball, a high precisionshaft-disk is fixed on the rotor. The actual motion of the rotor istested by arranging three measuring points on the surface of disk, twomeasuring points on the cylinder surface of shaft and installing anangle encoder on the shaft shoulder. The displacement sensorscorresponding to the five measuring points are fixed on the displacementsensor tooling. The scheme of the rotor motion test is developed basedon the principle of rigid body kinematics and the instantaneous six-DOFmotion of the rotor is decoupled by the discrete testing data ofdisplacement sensors and angle encoder. Therefore, the universality ofthe testing and precise decoupling of six-DOF motion parameters of anyrotors are settled which provides the guidance for precision equipmentaccuracy evaluation, error compensation and fault diagnosis.

The specific technical solution of the invention is as follow: The stepsof the six-DOF motion testing and motion parameter decoupling method forrotors based on shaft-disk are:

Step 1: A shaft-disk and a displacement sensor tooling are prepared, andthe shaft disk is an integral body including a standard shaft, a diskand a shaft shoulder. The standard shaft is located between the disk andthe shaft shoulder, whose axis is perpendicular to the disk surface andconcentric with those of the disk and shaft shoulder. The displacementsensor tooling includes the sensor mounting holes and threaded holes forthe installation of the displacement sensors. The tolerance of flatness,cylindricity, verticality and position of the shaft-disk anddisplacement sensor tooling should be one order of magnitude higher thanthe motion accuracy of the rotor.

Step 2: Two radial displacement sensors A₁ and A₂ are orthogonallyarranged on the cylindrical surface of the shaft, and three axialdisplacement sensors A₃, A₄ and A₅ are uniformly arranged on the surfaceof disk, and the angle encoder is installed at shaft shoulder. The fivedisplacement sensors are all fixed on the displacement sensor toolingwhich ensures the accuracy of measuring position of the displacementsensors.

Step 3: The non-measured surface of the disk is fixed on the rotor. Thefixed coordinate system S_(f){O_(f); X_(f), Y_(f), Z_(f)} ofdisplacement sensor tooling and the shaft-disk moving coordinate systemS_(m){O_(m); X_(m), Y_(m), Z_(m)} are established, of which the O_(f)and O_(m) are the center of displacement sensor tooling and shaft-diskrespectively, X_(f), Y_(f) and Z_(f) are parallel to the direction ofradial and axial displacement sensors respectively, O_(m)−X_(m), Y_(m)is coincident with the disk surface of the shaft-disk, and Z_(m) iscoincident with the axis of shaft. The coordinate axes of S_(f) andS_(m) are parallel at the initial moment.

Step 4: The six-DOF motion parameters of the rotor are described by thetranslational motion parameters (x,y,z) and rotational motion parameters(θ_(x),θ_(y),θ_(z)) in three directions of the axes of the disk-shaftcoordinate system S_(m) relative to the fixed coordinate system S_(f).The transformation relationship between the position vector r_(Pf) andr_(Pm) of any point P on the rotor in the fixed coordinate system andthe moving coordinate system is:r _(Pf) =r _(om) +r _(Pm) =r _(om) +R _(fm) r _(Pm)  (1)where, r_(om) is the translation transformation matrix,r_(om)=(x,y,z)^(T), R_(fm) is the rotation transformation matrix,

${R_{fm} = \begin{bmatrix}{c\;\theta_{y}c\;\theta_{z}} & {{s\;\theta_{x}s\;\theta_{y}c\;\theta_{z}} - {c\;\theta_{x}s\;\theta_{z}}} & {{c\;\theta_{x}s\;\theta_{y}c\;\theta_{z}} + {s\;\theta_{x}s\;\theta_{z}}} \\{c\;\theta_{y}s\;\theta_{z}} & {{s\;\theta_{x}s\;\theta_{y}s\;\theta_{z}} + {c\;\theta_{x}c\;\theta_{z}}} & {{c\;\theta_{x}s\;\theta_{y}s\;\theta_{z}} - {s\;\theta_{x}c\;\theta_{z}}} \\{{- s}\;\theta_{y}} & {s\;\theta_{x}c\;\theta_{y}} & {c\;\theta_{x}c\;\theta_{y}}\end{bmatrix}},$c and s are the abbreviations for cos and sin respectively.

Step 5: The position of the end points of the displacement sensors andthe measured values are represented by vectors, and the relationshipbetween the six-DOF parameters of the rotor and the measured values isestablished.

The measuring direction of radial displacement sensors A₁ and A₂intersects the standard shaft at points Q₁ and Q₂. Make the linesvertical to the axis of shaft through Q₁ and Q₂ which intersect the axisat P₁ and P₂ respectively. The closed loop vector equations of rigidbody kinematics is established for any motion position j of the rotor:

$\begin{matrix}\{ \begin{matrix}{{r_{Pi}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}} & {{{i = 1},2}\mspace{25mu}} \\{{r_{Om}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}} & {{i = 3},4,5} \\{{{d_{Ai}^{(j)}} = {d\text{/}2}}\mspace{110mu}} & {{{i = 1},2}\mspace{25mu}}\end{matrix}  & (2)\end{matrix}$where, r_(Ai) is the position vector of the end point of each sensorwhich is a known quantity, S_(Ai) ^((j)) is the vector from the endpoint of each displacement sensor to the measured point of the shaft ordisk surface which is the measured value, r_(Pi) ^((j))=R_(fm) ^((j))(0,0,z_(pi) ^((j)))^(T)+r_(Om) ^((j)) is the vector of P₁ or P₂ in thefixed coordinate system, R_(fm) ^((j)) is the rotation transformationmatrix including three rotational motion parameters, d_(Ai) ^((j)) isthe vector vertical to the Z_(m) in the moving coordinate system, r_(Om)^((j)) is the translation of the moving coordinate system relative tothe fixed coordinate system, including three translational motionparameters. There are 18 undetermined parameters including r_(Om)^((j)), R_(fm) ^((j)), z_(pi) ^((j)), d_(Ai) ^((j)), and 17 scalarequations. Combined with the condition of the rotation angle measured byangle encoder, the six-DOF motion parameters of the rotor including x,y, z, θ_(x), θ_(y) and θ_(z) can be decoupled, and the trajectory of anypoint or line of the rotor can be determined to evaluate the motionperformance of the rotor.

Step 5 above can be realized by another scheme: The position of the endpoints of the displacement sensors and the measured values arerepresented by vectors, and the relationship between the six-DOF motionparameters of the rotor and the measured values is established.

The measuring direction of radial displacement sensors A₁ and A₂intersects the standard shaft at points Q₁ and Q₂. Make the linesvertical to the axis of shaft through Q₁ and Q₂ which intersect the axisat P₁ and P₂ respectively. The closed loop vector equations of rigidbody kinematics is established for any motion position j of the rotor:

$\begin{matrix}\{ \begin{matrix}{{{r_{Pi}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}}\mspace{79mu}} & {{{i = 1},2}\mspace{70mu}} \\{{{r_{Om}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}}\mspace{76mu}} & {{{i = 3},4,5}\mspace{45mu}} \\{{( {r_{P\; 1}^{(j)} - r_{Om}^{(j)}} ) \times ( {r_{P\; 2}^{(j)} - r_{Om}^{(j)}} )} = 0} & \; \\{{{d_{Ai}^{(j)}} = {d\text{/}2}}\mspace{185mu}} & {{{i = 1},2}\mspace{70mu}} \\{{{d_{Ai}^{(j)} \cdot ( {r_{P\; 1}^{(j)} - r_{Om}^{(j)}} )} = 0}} & {{i = 1},2,\ldots\;,5}\end{matrix}  & (3)\end{matrix}$where, r_(Ai) is the position vector of the end point of each sensorwhich is a known quantity, S_(Ai) ^((j)) is the vector from the endpoint of each displacement sensor to the measured point of shaft or disksurface which is the measured value, r_(P1) ^((j)), r_(P2) ^((j)) andr_(Om) ^((j)) are the vector of P₁, P₂ and O_(m) from the origin O_(f)of the fixed coordinate system, d_(Ai) ^((j)) is the vector from P_(i)or O_(m) to Q_(i). There are 24 undetermined parameters including r_(P1)^((j)), r_(P2) ^((j)), r_(Om) ^((j)), d_(Ai) ^((j)), and 24 scalarequations to solve the direction vector of the axis k_(m)=(r_(P1)^((j))−r_(Om) ^((j)))/|r_(P1) ^((j))−r_(Om) ^((j))|. The directionvector of the rotor axis can be obtained without the condition of therotation angle measured by angle encoder, including x, y, z, θ_(x) andθ_(y). Combined with the rotation angle θ_(z) measured by angle encoder,the trajectory of any point or line of the rotor can be determined toevaluate the motion performance of the rotor.

The beneficial effect of the invention is that the rotor motion istested by using the shaft-disk, which is convenient for arranging thedisplacement sensors and reducing the space occupied by the motion test.Based on the principle of rigid body kinematics, the accurate decouplingproblem of the six-DOF motion parameters of the rotor can be solved.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of the shaft-disk in a specificimplementation of the invention.

FIG. 2 is a schematic diagram of the displacement sensor tooling in aspecific implementation of the invention.

FIG. 3 is a schematic diagram of the measuring point layout ofshaft-disk in a specific implementation of the invention.

FIG. 4 is a schematic diagram of the coordinate system definition andclosed-loop vector in a specific implementation of the invention.

FIG. 5 is a diagram of the decoupling curves in a specificimplementation of the invention.

FIG. 6 is a schematic diagram of the rotor motion test structure in aspecific implementation of the invention.

In the figures, 1 is the surface of disk, 2 is the cylindrical surfaceof standard shaft, 3 is the shaft shoulder, 4 is the displacementsensor, 5 is the clamping screw, 6 is the displacement sensor tooling, 7is the shaft-disk and 8 is the angle encoder.

DETAILED DESCRIPTION

In order to explain the technical solution of the invention explicitly,the invention is further described in combination with the attachedfigures and specific implementation cases.

The object of this embodiment is to test the six-DOF motion of a rotorand decouple the six-DOF motion parameters through the shaft-disk,displacement sensors and angle encoder.

The schematic diagram of relevant parameters of rotor motion test isshown in FIG. 3, and the parameter values are shown in Table 1.

TABLE 1 Parameters of the rotor motion test scheme d D l θ₁ θ₂ θ₃ 33 mm114 mm 29 mm 30° 120° 120°

Upon the six-DOF motion testing and decoupling method for rotors, thespecific implementation steps are as follow:

Step 1: A shaft-disk and a displacement sensor tooling are prepared asshown in FIG. 1 and FIG. 2 meeting the parameters in Table 1. The shaftdisk is an integral body including a standard shaft, a disk and a shaftshoulder. The standard shaft is located between the disk and the shaftshoulder, whose axis is perpendicular to the disk surface and concentricwith those of the disk and shaft shoulder. The displacement sensortooling includes the sensor mounting holes and threaded holes for theinstallation of the displacement sensors. The tolerance of flatness,cylindricity, verticality and position of the shaft-disk anddisplacement sensor tooling should be one order of magnitude higher thanthe motion accuracy of the rotor.

Step 2: Two radial displacement sensors A₁ and A₂ are orthogonallyarranged on the cylindrical surface of the shaft, and three axialdisplacement sensors A₃, A₄ and A₅ are uniformly arranged on the surfaceof disk, and the angle encoder is installed at shaft shoulder. The fivedisplacement sensors are all fixed on the displacement sensor toolingwhich ensures the accuracy of measuring position of the displacementsensors. The testing scheme is shown in FIG. 3.

Step 3: The non-measured surface of the disk is fixed on the rotor. Thefixed coordinate system S_(f){O_(f); X_(f), Y_(f), Z_(f)} ofdisplacement sensor tooling and the shaft-disk moving coordinate systemS_(m){O_(m); X_(m), Y_(m), Z_(m)} are established of which the O_(f) andO_(m) are the center of displacement sensor tooling and shaft-diskrespectively, X_(f), Y_(f) and Z_(f) are parallel to the direction ofradial and axial displacement sensors respectively, O_(m)−X_(m)Y_(m) iscoincident with the disk surface of the shaft-disk, and Z_(m) iscoincident with the axis of shaft. The coordinate axes of S_(f) andS_(m) are parallel at the initial moment. The definition of coordinatesystem is shown in FIG. 4.

Step 4: The six-DOF motion parameters of the rotor are described by thetranslational motion parameters (x,y,z) and rotational motion parameters(θ_(x),θ_(y),θ_(z)) in three directions of the axes of the disk-shaftcoordinate system S_(m) relative to the fixed coordinate system S_(f).The transformation relationship is shown in Equation (1).

Step 5: The position of the end points of the displacement sensors andthe measured values are represented by vectors. The end points of eachdisplacement sensor are r_(A1)=(0,16.5,0)^(T), r_(A2)=(−16.5,0,0)^(T),r_(A3)=(0,−57,−29)^(T), r_(A4)=(−49.36,28.5, −29)^(T) andr_(A5)=(49.36,28.5,−29)^(T). Record the values of each displacementsensor during the process of the rotor motion, and the measured vectorsare S_(A1) ^((j))=(0,−S_(A1) ^((j)),0)^(T), S_(A2) ^((j))=(S_(A2)^((j)), 0,0)^(T), S_(A3) ^((j))=(0, 0,−S_(A3) ^((j)))^(T), S_(A4)^((j))=(0, 0,−S_(A4) ^((j)))^(T) and S_(A5) ^((j))=(0, 0,−S_(A5)^((j)))^(T), where S_(Ai) ^((j)), i=1, 2, . . . , 5 are the measuredvalues of each displacement sensor. According to the Equation (2) or(3), and the discrete measurement data of displacement sensors and angleencoder, the six-DOF motion parameters of the rotor including x, y, z,θ_(x), θ_(y) and θ_(z) can be decoupled as shown in FIG. 5.

The invention claimed is:
 1. A six-DOF motion testing and motionparameter decoupling method for rotors based on shaft-disk, whereincomprising the following steps: step 1: a shaft-disk and a displacementsensor tooling are prepared, and the shaft disk is an integral bodyincluding a standard shaft, a disk and a shaft shoulder; the standardshaft is located between the disk and the shaft shoulder, whose axis isperpendicular to the disk surface and concentric with those of the diskand shaft shoulder; the displacement sensor tooling includes sensormounting holes and threaded holes for the installation of thedisplacement sensors; the tolerance of flatness, cylindricity,verticality and position of the shaft-disk and displacement sensortooling should be one order of magnitude higher than the motion accuracyof the rotor; step 2: two radial displacement sensors A₁ and A₂ areorthogonally arranged on the cylindrical surface of the shaft, and threeaxial displacement sensors A₃, A₄ and A₅ are uniformly arranged on thesurface of disk, and the angle encoder is installed at shaft shoulder;the five displacement sensors are all fixed on the displacement sensortooling which ensures the accuracy of measuring position of thedisplacement sensors; step 3: the non-measured surface of the disk isfixed on the rotor; the fixed coordinate system S_(f){O_(f); X_(f),Y_(f), Z_(f)} of displacement sensor tooling and the shaft-disk movingcoordinate system S_(m){O_(m); X_(m), Y_(m), Z_(m)} are established, ofwhich the O_(f) and O_(m) are the center of displacement sensor toolingand shaft-disk respectively, X_(f), Y_(f) and Z_(f) are parallel to thedirection of radial and axial displacement sensors respectively,O_(m)−X_(m)Y_(m) is coincident with the disk surface of the shaft-disk,and Z_(m) is coincident with the axis of shaft; the coordinate axes ofS_(f) and S_(m) are parallel at the initial moment; step 4: the six-DOFmotion parameters of the rotor are described by the translational motionparameters (x,y,z) and rotational motion parameters (θ_(x),θ_(y),θ_(z))in three directions of the axes of the disk-shaft coordinate systemS_(m) relative to the fixed coordinate system S_(f), the transformationrelationship between the position vector r_(pf) and r_(pm) of any pointP on the rotor in the fixed coordinate system and the moving coordinatesystem is:r _(Pf) =r _(om) +r _(Pm) =r _(om) +R _(fm) r _(Pm)  (1) where, r_(om)is the translation transformation matrix, r_(om)=(x,y,z)^(T), R_(fm) isthe rotation transformation ${R_{fm} = \begin{bmatrix}{c\;\theta_{y}c\;\theta_{z}} & {{s\;\theta_{x}s\;\theta_{y}c\;\theta_{z}} - {c\;\theta_{x}s\;\theta_{z}}} & {{c\;\theta_{x}s\;\theta_{y}c\;\theta_{z}} + {s\;\theta_{x}s\;\theta_{z}}} \\{c\;\theta_{y}s\;\theta_{z}} & {{s\;\theta_{x}s\;\theta_{y}s\;\theta_{z}} + {c\;\theta_{x}c\;\theta_{z}}} & {{c\;\theta_{x}s\;\theta_{y}s\;\theta_{z}} - {s\;\theta_{x}c\;\theta_{z}}} \\{{- s}\;\theta_{y}} & {s\;\theta_{x}c\;\theta_{y}} & {c\;\theta_{x}c\;\theta_{y}}\end{bmatrix}},$ matrix, c and s are the abbreviations for cos and sinrespectively; step 5: the position of the end points of the displacementsensors and the measured values are represented by vectors, and therelationship between the six-DOF motion parameters of the rotor and themeasured values is established; the measuring direction of radialdisplacement sensors A₁ and A₂ intersects the standard shaft at pointsQ₁ and Q₂; make the lines vertical to the axis of shaft through Q₁ andQ₂ which intersect the axis at P₁ and P₂ respectively; the closed loopvector equations of rigid body kinematics is established for any motionposition j of the rotor: $\begin{matrix}\{ \begin{matrix}{{r_{Pi}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}} & {{{i = 1},2}\mspace{25mu}} \\{{r_{Om}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}} & {{i = 3},4,5} \\{{{d_{Ai}^{(j)}} = {d\text{/}2}}\mspace{110mu}} & {{{i = 1},2}\mspace{25mu}}\end{matrix}  & (2)\end{matrix}$ where, d is the diameter of the standard shaft r_(Ai) theposition vector of the end point of each sensor which is a knownquantity, S_(Ai) ^((j)) is the vector from the end point of eachdisplacement sensor to the measured point of the shaft or disk surfacewhich is the measured value, r_(Pi) ^((j))=R_(fm) ^((j)) (0,0,z_(pi)^((j)))^(T)+r_(Om) ^((j)) is the vector of P₁ or P₂ in the fixedcoordinate system, R_(fm) ^((j)) is the rotation transformation matrixincluding three rotational motion parameters, d_(Ai) ^((j)) is thevector vertical to the Z_(m) in the moving coordinate system, r_(Om)^((j)) is the translation of the moving coordinate system relative tothe fixed coordinate system, including three translational motionparameters; there are 18 undetermined parameters including r_(Om)^((j)), R_(fm) ^((j)), z_(pi) ^((j)), d_(Ai) ^((j)), and 17 scalarequations; combined with the condition of the rotation angle measured byangle encoder, the six-DOF motion parameters of the rotor including x,y, z, θ_(x),θ_(y) and θ_(z) can be decoupled, and the trajectory of anypoint or line of the rotor can be determined to evaluate the motionperformance of the rotor.
 2. The six-DOF motion testing and motionparameter decoupling method for rotor based on shaft-disk according toclaim 1, wherein, step 5: the position of the end points of thedisplacement sensors and the measured values are represented by vectors,and the relationship between the six-DOF motion parameters of the rotorand the measured values is established; the measuring direction ofradial displacement sensors A₁ and A₂ intersects the standard shaft atpoints Q_(i) and Q₂; make the lines vertical to the axis of shaftthrough Q₁ and Q₂ which intersect the axis at P₁ and P₂ respectively;the closed loop vector equations of rigid body kinematics is establishedfor any motion position j of the rotor: $\begin{matrix}\{ \begin{matrix}{{{r_{Pi}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}}\mspace{79mu}} & {{{i = 1},2}\mspace{70mu}} \\{{{r_{Om}^{(j)} + d_{Ai}^{(j)}} = {r_{Ai} + S_{Ai}^{(j)}}}\mspace{76mu}} & {{{i = 3},4,5}\mspace{45mu}} \\{{( {r_{P\; 1}^{(j)} - r_{Om}^{(j)}} ) \times ( {r_{P\; 2}^{(j)} - r_{Om}^{(j)}} )} = 0} & \; \\{{{d_{Ai}^{(j)}} = {d\text{/}2}}\mspace{185mu}} & {{{i = 1},2}\mspace{70mu}} \\{{{d_{Ai}^{(j)} \cdot ( {r_{P\; 1}^{(j)} - r_{Om}^{(j)}} )} = 0}} & {{i = 1},2,\ldots\;,5}\end{matrix}  & (3)\end{matrix}$ where, d is the diameter of the standard shaft, r_(Ai), isthe position vector of the end point of each sensor which is a knownquantity, S_(Ai) ^((j)) is the vector from the end point of eachdisplacement sensor to the measured point of shaft or disk surface whichis the measured value, r_(P1) ^((j)), r_(P2) ^((j)) and r_(Om) ^((j))are the vector of P₁, P₂ and O_(m) from the origin O_(f) of the fixedcoordinate system, d_(Ai) ^((j)) is the vector from P_(i) or O_(m) toQ_(i); there are 24 undetermined parameters including r_(P1) ^((j)),r_(P2) ^((j)), r_(Om) ^((j)), d_(Ai) ^((j)), and 24 scalar equations tosolve the direction vector of the axis k_(m)=(r_(P1) ^((j))−r_(Om)^((j)))/|r_(P1) ^((j))−r_(Om) ^((j))|; the direction vector of the rotoraxis can be obtained without the condition of the rotation anglemeasured by angle encoder, including x, y, z, θ_(x) and θ_(y), combinedwith the rotation angle θ, measured by angle encoder, the trajectory ofany point or line of the rotor can be determined to evaluate the motionperformance of the rotor.